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Question Number 106503 by Dwaipayan Shikari last updated on 05/Aug/20

$$\frac{2}{3}+\frac{2}{18}+\frac{2}{27}+\frac{2}{324}+....$$

Commented by Dwaipayan Shikari last updated on 05/Aug/20

$$2(\frac{1}{2+1}+\frac{1}{2{(2+1)}^2}+\frac{1}{3.{(2+1)}^3}+....)$$$$=2(\frac{1}{x}+\frac{1}{2x^2}+\frac{1}{3x^3}+....)$$$$=-2log(1-\frac{1}{x})=-2log\left(\frac{2}{3}\right)=log\left(\frac{9}{4}\right)x=3$$$$=0.81093..$$

Commented by Ar Brandon last updated on 05/Aug/20

I think you changed the question.�� I struggled hard to establish a general term.

Commented by mr W last updated on 05/Aug/20

$$2+\frac{2}{11}+\frac{2}{25}+\frac{2}{69}+...=?$$

Commented by Dwaipayan Shikari last updated on 05/Aug/20

Oh I had changed the question. Sorry for the inconvenience ��

Commented by Ar Brandon last updated on 05/Aug/20

��Mr W

Commented by Dwaipayan Shikari last updated on 05/Aug/20

$$Greatquestion$$

Commented by mr W last updated on 05/Aug/20

$$iwasjustwonderinghowyoucanfind$$$$thegeneraltermthroughjustfour$$$$giventerms!formeitisalmost$$$$impossibletoseethegeneralterm,$$$$forexamplemyexampleabove.$$

Commented by prakash jain last updated on 05/Aug/20

$$\mathrm{Suppose}1..k\mathrm{terms}\mathrm{of}\mathrm{a}\mathrm{sequence}$$$$\mathrm{are}\mathrm{given}\mathrm{and}\mathrm{formula}$$$$g\left(n\right)\mathrm{satisifies}\mathrm{the}\mathrm{given}\mathrm{terms}.$$$$\mathrm{then}f\left(n\right)=h\left(n\right)\underset{j=1}{\prod ^k}(n-k)+g\left(n\right)$$$$\mathrm{will}\mathrm{also}\mathrm{satify}\mathrm{the}\mathrm{given}\mathrm{sequence}.$$$$\mathrm{where}h\left(n\right)\mathrm{is}\mathrm{any}\mathrm{arbitary}\mathrm{function}$$$$\mathrm{on}n.$$$$\mathrm{So}\mathrm{there}\mathrm{always}\mathrm{exists}\mathrm{infinite}\mathrm{number}$$$$\mathrm{of}\mathrm{formula}\mathrm{for}\mathrm{a}\mathrm{given}\mathrm{finite}\mathrm{terms}.$$

Commented by Ar Brandon last updated on 06/Aug/20

��

Commented by 1549442205PVT last updated on 06/Aug/20

$$\mathrm{We}\mathrm{can}\mathrm{find}\mathrm{the}\mathrm{general}\mathrm{term}\mathrm{in}\mathrm{form}$$$$\frac{2}{{\mathrm{a}}_{\mathrm{n}}}\mathrm{with}{\mathrm{a}}_{\mathrm{n}}=\mathrm{f}\left(\mathrm{x}\right)={\mathrm{ax}}^3+{\mathrm{bx}}^2+\mathrm{cx}+\mathrm{d}\mathrm{with}\mathrm{condition}$$$$\mathrm{that}\mathrm{f}\left(1\right)=1,\mathrm{f}\left(2\right)=11,\mathrm{f}\left(3\right)=25,\mathrm{f}\left(4\right)=69$$$$\mathrm{which}\mathrm{lead}\mathrm{to}\mathrm{solve}\mathrm{the}\mathrm{system}\mathrm{of}\mathrm{four}$$$$\mathrm{unknows}\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\mathrm{and}\mathrm{get}$$$$\mathrm{a}=\frac{13}{3},\mathrm{b}=-24,\mathrm{c}=\frac{155}{3},\mathrm{d}=-31\mathrm{and}\mathrm{so}$$$${\mathrm{a}}_{\mathrm{n}}=\frac{13}{3}{\mathrm{n}}^3-{24\mathrm{n}}^2+155\mathrm{n}-31\mathrm{which}\mathrm{from}$$$$\mathrm{this}\mathrm{we}\mathrm{get}\mathrm{the}\mathrm{next}\mathrm{terms}\mathrm{of}\mathrm{the}\mathrm{given}$$$$\mathrm{sequence}:\frac{2}{169},\frac{2}{351},\frac{2}{641},\frac{2}{1065},...\mathrm{and}$$$$\mathrm{Of}\mathrm{course},\mathrm{here}\mathrm{need}\mathrm{must}\mathrm{be}\mathrm{suppose}$$$$\mathrm{that}\mathrm{any}\mathrm{formular}\mathrm{was}\mathrm{found}\mathrm{out}\mathrm{only}$$$$\mathrm{need}\mathrm{to}\mathrm{correct}\mathrm{for}\mathrm{four}\mathrm{given}\mathrm{terms}$$$$\mathrm{because}\mathrm{otherwise}\mathrm{we}\mathrm{can}\mathrm{find}\mathrm{a}$$$$\mathrm{polynomial}\mathrm{of}\mathrm{degree}5\mathrm{which}\mathrm{also}\mathrm{get}$$$$\mathrm{four}\mathrm{first}\mathrm{values}\mathrm{of}\mathrm{given}\mathrm{sequence}$$$$\mathrm{Which}\mathrm{this}\mathrm{completely}\mathrm{depend}\mathrm{on}\mathrm{the}$$$$\mathrm{viewer}\mathrm{pointof}\mathrm{person}\mathrm{give}\mathrm{the}\mathrm{question}$$$$$$

Commented by Ar Brandon last updated on 06/Aug/20

Wow, thanks for the knowledge Sir.

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